To troubleshoot your project, you could, stepwise:
1/ Connect the RTC device alone to the DUE and debug until it works
http://forum.arduino.cc/index.php?topic=168977.0
2/ Connect the VS1053B device alone to the DUE and debug until it works
Since the DS3231 communicates via I2C and the VS1053B via SPI, unless you have somewhere a blocking code, they should work together on the DUE.